31edo: Difference between revisions

Line 76:

|}

Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning supporting [[meantone]], but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of 7/4, a cent flat, is also very close to just. Because of these near-just values and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31-et is relatively quite accurate and is [[The Riemann Zeta Function and Tuning#Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap edo and a zeta peak edo]], and represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is consistent through, making it a very tone-efficient melodic approximation of the [[11-limit]], although the 14/11 and 9/7, which are equated, may be too off for some. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course).

Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning supporting [[meantone]], but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of 7/4, a cent flat, is also very close to just. Because of these near-just values and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31-et is relatively quite accurate and is [[The Riemann Zeta Function and Tuning#Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap edo and a zeta peak edo]].

31edo's 5\31 neutral third generator generates [[mosh]] and [[dicoid]] MOSes. Its 12\31 generator generates an [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8 and 5/4 better in tune.

~~31edo is the 11th [[prime numbers|prime]] edo, following [[29edo]] and coming before [[37edo~~]].

== Intervals ==

@@ Line 76: / Line 76: @@
 |}
-Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning supporting [[meantone]], but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of 7/4, a cent flat, is also very close to just. Because of these near-just values and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31-et is relatively quite accurate and is [[The Riemann Zeta Function and Tuning#Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap edo and a zeta peak edo]], and represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is consistent through, making it a very tone-efficient melodic approximation of the [[11-limit]], although the 14/11 and 9/7, which are equated<!-- (thus tempering [[99/98]])-->, may be too off for some. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). <!--In the [[13-limit]] it doesn't do as well, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.-->
+Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning supporting [[meantone]], but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of 7/4, a cent flat, is also very close to just. Because of these near-just values and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31-et is relatively quite accurate and is [[The Riemann Zeta Function and Tuning#Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap edo and a zeta peak edo]].
-edo's 5\31 neutral third generator generates [[mosh]] and [[dicoid]] MOSes. Its 12\31 generator generates an [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8 and 5/4 better in tune.
-edo is the 11th [[prime numbers|prime]] edo, following [[29edo]] and coming before [[37edo]].
 == Intervals ==